A character has access to a roc egg. I would like to work out at what point it could be ridden, and when it stops being a cute bird and turns into a man-eating monster that locals might object to having arrive with the party. How quickly would it grow?
Depends on what you want to compare it to I'd think
I had a look at the Emu (as one of the largest birds alive today) and their chicks are full-grown within 5-6 months, they are still looked after their father during this time (up to seven months) so could be considered "young" during this stage. They can leave the nest within a few days so they quickly become semi-independant.
Given the size of a roc this may stretch out a bit more; but even so within a year the party are going to have themselves a massive beast that's easily capable of carrying a character, and since most animals breed once a year that's going to be the longest marker for age.
So going from this; I'd roughly extropolate:
Cute and fluffy chicks: Up to a week. Awww, it ate your finger.
Make sure they keep it well fed!
One approach would be to ask yourself how quickly does the story require it to grow? If it isn't explicitly described in the 50 Fathoms rules, then my feeling is that it would be open to interpretation, and that you should do whatever is needed to make an interesting adventure for the Roc's owner.
This actually doesn't have an answer! Here's some background explaining why:
That means it's up to you! There are a few ways you can decide this.
So, how long it takes hatchling rocs to mature into young adults could take anywhere from "not applicable" (for the unique Roc) to 20+ years, and anywhere in between.